arXiv subjects

Daniel Allcock

Publications and source records attributed to Daniel Allcock.

At least 37 records · Page 2Linked to original sources

arXiv2015

Presentation of affine Kac-Moody groups over rings

Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits-style presentation for the Steinberg group St of any rank > 2 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1- and 2-node subdiagrams of the Dynkin diagram. This leads to a completely explicit presentation. Using this we show that St is finitely presented if the rank is > 3 and R is finitely generated as a ring, or if the rank is 3 and R is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac-Moody groups when R is a Dedekind domain of arithmetic type.

math.GR↗

arXiv2015

Geometric generators for braid-like groups

We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n dimensional complex vector space, or n dimensional complex hyperbolic space, or the Hermitian symmetric space for O(2,n), and then takes the quotient by a discrete group $PΓ$. The classical example is the braid group, but there are many similar "braid-like" groups that arise in topology and algebraic geometry. Our main result is that if $PΓ$ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then G is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for G in a particular intricate example in complex hyperbolic space of dimension 13. The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group M, that gives geometric meaning to the generators and relations in the Conway-Simons presentation of $(M \times M):2$.

math.GT↗

arXiv2014

A fake plane via 2-adic uniformization with torsion

We adapt the theory of non-Archimedean uniformization to construct a smooth surface from a lattice in PGL3(Q2) that has nontrivial torsion. It turns out to be a fake projective plane, commensurable with Mumford's fake plane yet distinct from it and the other fake planes that arise from 2-adic uniformization by torsion-free groups. As part of the proof, and of independent interest, we compute the homotopy type of the Berkovich space of our plane.

math.AG↗

arXiv2014

Root systems for Lorentzian Kac-Moody algebras in rank 3 (arXiv version)

Sometimes a hyperbolic Kac-Moody algebra admits an automorphic correction, meaning a generalized Kac-Moody algebra with the same real simple roots and whose denominator function has good automorphic properties; these for example allow one to work out the root multiplicities. Gritsenko and Nikulin have formalized this in their theory of Lorentzian Lie algebras and shown that the real simple roots must satisfy certain conditions in order for the algebra to admit an automorphic correction. We classify the hyperbolic root systems of rank 3 that satisfy their conditions and have only finite many simple roots, or equivalently a timelike Weyl vector. There are 994 of them, with as many as 24 simple roots. Patterns in the data suggest that some of the non-obvious cases may be the richest.

math.RT↗

arXiv2013

Reflection centralizers in Coxeter groups

We refine Brink's theorem, that the non-reflection part of a reflection centralizer in a Coxeter group W is a free group. We give an explicit set of generators for centralizer, which is finitely generated when W is. And we give a method for computing the Coxeter diagram for its reflection subgroup. In many cases, our method allows one to compute centralizers in one's head.

math.GR↗

arXiv2011

Unabridged table of reflective lattices of rank 3

This is the unabridged table of all 8595 rank three reflective Lorentzian lattices, intended as a supplement to the author's paper classifying them. The abridged table in that paper is complete too, but less explicit, so the main purpose of this document is archival. The TeX sourcecode is simultaneously a Perl script, which when run prints out all the lattices in computer-readable format.

math.GR↗

arXiv2011

Completions, branched covers, Artin groups and singularity theory

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT(k) inequality. We prove a general CAT(k) extension theorem, giving sufficient conditions on and near the boundary of a locally CAT(k) metric space for the completion to be CAT(k). We use this to prove that a branched cover of a complete Riemannian manifold is locally CAT(k) if and only if all tangent spaces are CAT(0) and the base has sectional curvature bounded above by k. We also show that the branched cover is a geodesic space. Using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally CAT(k) and the complement of the branch locus to be contractible. We conjecture that the universal branched cover of complex Euclidean n-space over the mirrors of a finite Coxeter group is CAT(0). Conditionally on this conjecture, we use our machinery to prove the Arnol'd-Pham-Thom conjecture on K(pi,1) spaces for Artin groups. Also conditionally, we prove the asphericity of moduli spaces of amply lattice-polarized K3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in Arnol'd's hierarchy.

math.GT↗

arXiv2011

Normalizers of Parabolic Subgroups of Coxeter Groups

We improve a bound of Borcherds on the virtual cohomological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink's result that the non-reflection part of a reflection centralizer is free. Namely, the non-reflection part of the normalizer of parabolic subgroup of type D5 or Aodd is either free or has a free subgroup of index 2.

math.GR↗

arXiv2010

The reflective Lorentzian lattices of rank 3

We classify all the symmetric integer bilinear forms of signature (2,1) whose isometry groups are generated up to finite index by reflections. There are 8595 of them up to scale, whose 374 distinct Weyl groups fall into 39 commensurability classes. This extends Nikulin's enumeration of the strongly square-free cases. Our technique is an analysis of the shape of the Weyl chamber, followed by computer work using Vinberg's algorithm and our "method of bijections". We also correct a minor error in Conway and Sloane's definition of their canonical 2-adic symbol.

math.GR↗

arXiv2010

Triangles of Baumslag-Solitar Groups

Our main result is that many triangles of Baumslag-Solitar groups collapse to finite groups, generalizing a famous example of Hirsch. A triangle of Baumslag-Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of class<4. We obtain a lot of information about finite quotients, even when we cannot determine developability.

math.GR↗

arXiv2009

Hyperbolic geometry and moduli of real cubic surfaces

Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4 and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in PO(4,1). We also derive several new and several old results on the topology of M_0^R. Let M_s^R be the moduli space of real cubic surfaces that are stable in the sense of geometric invariant theory. We show that this space carries a hyperbolic structure whose restriction to M_0^R is that just mentioned. The corresponding lattice in PO(4,1), for which we find an explicit fundamental domain, is nonarithmetic.

math.AG↗

arXiv2009

Infinitely many hyperbolic Coxeter groups through dimension 19

We prove the following: there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space H^n for every n < 20 (resp. n < 7). When n=7 or 8, they may be taken to be nonarithmetic. Furthermore, for 1 < n < 20, with the possible exceptions n=16 and 17, the number of essentially distinct Coxeter groups in H^n with noncompact fundamental domain of volume less than or equal to V grows at least exponentially with respect to V. The same result holds for cocompact groups for n < 7. The technique is a doubling trick and variations on it; getting the most out of the method requires some work with the Leech lattice.

math.GR↗

arXiv2008

A Banach space determined by the Weil height

The absolute logarithmic Weil height is well defined on the group of units of the algebraic closure of the rational numbers, modulo roots of unity, and induces a metric topology on this group. We show that the completion of this metric space is a Banach space over the field of real numbers. We further show that this Banach space is isometrically isomorphic to a co-dimension one subspace of L1 of a certain totally disconnected, locally compact space, equipped with a certain measure satisfying an invariance property with respect to the absolute Galois group.

math.NT↗

arXiv2008

Monodromy Groups of Hurwitz-type Problems

We solve the Hurwitz monodromy problem for degree-4 covers. That is, the Hurwitz space H_{4,g} of all simply branched covers of P^1 of degree 4 and genus g is an unramified cover of the space P_{2g+6} of (2g+6)-tuples of distinct points in P^1. We determine the monodromy of pi_1(P_{2g+6}) on the points of the fiber. This turns out to be the same problem as the action of pi_1(P_{2g+6}) on a certain local system of Z/2-vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, gcd(3,N)=1, in place of Z/2. This in turn allows us to answer a question of Ellenberg concerning families of Galois covers of P^1 with deck group (Z/N)^2:S_3.

math.GR↗

arXiv2008

On the Y555 complex reflection group

We give a computer-free proof of a theorem of Basak, describing the group generated by 16 complex reflections of order 3, satisfying the braid and commutation relations of the Y555 diagram. The group is the full isometry group of a certain lattice of signature (13,1) over the Eisenstein integers Z[cube root of 1]. Along the way we enumerate the cusps of this lattice and classify the root and Niemeier lattices over this ring.

math.GR↗

arXiv2007

Linear Algebraic Groups without the Normalizer Theorem

One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps in the standard development, including the self-normalizing property of Borel subgroups.

math.RT↗

arXiv2007

A Monstrous Proposal

We explain a conjecture relating the monster simple group to an algebraic variety that was discovered in a non-monstrous context.

math.GR↗

arXiv2006

The Moduli Space of Cubic Threefolds as a Ball Quotient

The moduli space of cubic threefolds in CP4, with some minor birational modifications, is the Baily-Borel compactification of the quotient of the complex 10-ball by a discrete group. We describe both the birational modifications and the discrete group explicitly.

math.AG↗